CHAPTER 3 CONVENTIONAL TERRESTRIAL REFERENCE FRAME Definition The Terrestrial Reference System adopted for either the analysis of individual data sets by techniques (VLBI, SLR, LLR, GPS...) or the combination of individual Solutions into a unified set of data (Station coordinates, Earth orientation parameters, etc..) follows these criteria (Boucher, 1991): a) It is geocentric, the center of mass being defined for the whole Earth, including oceans and atmosphere. b) Its scale is that of a local Earth frame, in the meaning of a relativistic theory of gravitation. c) Its orientation is given by the BIH orientation at 1984.0. d) Its time evolution in orientation will create no residual global rotation with regards to the crust. Realization When one wants to realize such a conventional terrestrial reference system through a reference frame i.e. a network of Station coordinates, it will be specified by Cartesian equatorial coordinates X, Y, and Z, by preference. If geographical coordi­ nates are needed, the GRS80 ellipsoid is recommended (see table 3.2) . Each analysis center compares its reference frame to a realization of the Conventional Terrestrial Reference System (CTRS), as described above. Within IERS, each Terrestrial Reference System (TRF) is either directly, or after transformation, expressed as a realization of the CTRS adopted by IERS as its ITRS. The position of a point located on the surface of the solid Earth should be expressed by X(t) = X0 + VQ(t-t0) +£AXi(ü), 2 where AXj are corrections to various time changing effects, and X0 and V0 are position and velocity at the epoch t0. The corrections to be considered are solid Earth tide displacement (füll correction including permanent effect, so that the extra correction which was originally recommended in order to have zero mean correction is no longer valid), ocean loading, and atmospheric loading. 16 Further corrections could be added if they are at mm level and can be computed by a suitable model. The velocity V0 should be expressed as where VpUtc is the horizontal velocity computed from the NNR-NUVEL-1 model (DeMets, et al. , 1990; Argus and Gordon, 1991) and Vr a residual velocity. In data analysis, X0 and Vr should be considered as solve-for Parameters. In particular, if a non linear change occurs (earth- quake, volcanic event ...), a new X0 parameter should be adopted. When adjusting parameters, particularly velocities, the IERS orientation should be kept at all epochs, ensuring the alignment at a reference epoch and the time evolution through a no net rotation condition. The way followed by various analysis centers depends on their own view of modelling, and on the techniques themselves. For the origin, only data which can be modelled by dynamical techniques (currently SLR, LLR or GPS for IERS) can determine the center of mass. The VLBI system can be referred to a geocentric system by adopting for a Station its geocentric position at a reference epoch as provided from external Information. The scale is obtained by appropriate relativistic modelling. This is particularly true for VLBI and LLR which are usually modelled in a barycentric frame. A more detailed treatment can be found in chapter 14. The orientation is defined by adopting IERS (or BIH) Earth orientation parameters at a reference epoch. In the case of SLR, an additional constraint in longitude is necessary. The unit of length is the meter (SI) . The IERS Reference Pole (IRP) and Reference Meridian (IRM) are consistent with the corresponding directions in the BIH Terrestrial System (BTS) within ± 07 005. The BIH reference pole was adjusted to the Conventional International Origin (CIO) in 1967; it was then kept stable independently until 1987. The uncertainty of the tie of the IRP with the CIO is ± 0703. The time evolution of the orientation will be insured by using a no-net-rotation condition with regards to horizontal tectonic motions over the whole Earth. Transformation Parameters of World Coordinate Systems and Datums The seven-parameter (similarity transformation between any two Cartesian Systems, e. g., from (u,v,w) to (x,y,z), or in short (u,v,w) -* (x,y,z) can be written (Soler and Hothem, 1989) as 17 X AX I 1 5w -50 y Ay > + (1 + <Ss) -c?o> 1 5€ (1) z AZ 50 -5e 1 where Ax,Ay,Az = coordinates of the origin of the frame (u,v,w) in the frame (x,y,z); Se, 50, So> = differential rotations (expressed in radians) respectively, around the axes (u,v,w) to establish parallelism with the (x,y,z) frame (Positive rotations are counterclockwise rotations as viewed looking tov/ard the origin of the right handed coordinate system.); and 5 s = differential scale change (expressed in ppm X 10"6) (see Table 3.1). Table 3.1. Transformation parameters from ITRF90 to major global or local datums. Coordinate Ax Ay Az 6s 5c 60 6o) System Tl T2 T3 D -Rl -R2 -R3 (datum) (m) (m) (m) (ppm) (") C) WGS84 0.060 -0.517 -0.223 -0.011 -0.0183 0.0003 -0.0070 WGS72 0.060 -0.517 -4.723 -0.231 -0.0183 0.0003 0.5470 NWL9D 0.060 -0.517 -4.723 0.599 -0.0183 0.0003 0.8070 BTS84 -0.058 0.028 -0.036 0.030 -0.0035 -0.0020 -0.0017 BTS85 -0.004 0.049 0.006 0.025 -0.0026 0.0005 0.0014 BTS86 0.027 -0.011 -0.044 0.008 -0.0008 0.0023 0.0072 BTS87 -0.011 -0.008 -0.057 0.006 -0.0004 -0.0002 -0.0003 ITRFO -0.007 -0.009 -0.055 0.005 -0.0004 -0.0002 -0.0001 ITRF88 0.000 -0.012 -0.062 0.006 -0.0001 0.0000 0.0000 ITRF89 0.005 0.024 -0.038 0.003 0.0000 0.0000 0.0000 Once the Cartesian coordinates (x,y,z) are known, they can be transformed to "datum11 or curvilinear geodetic coordinates (A,0,h) referred to an ellipsoid of semi-major axis a and flattening f (see Table 3.2), using the following noniterative method (Bowring, 1985): tan X (2) 18 2 tan <p = z(l - f) + e a sin'u (3) (1 - f) (p - e^a COSJ/LI) (4) h = p cos 0 + z sin <p - a(l - e7sin2<p)v', (5) e2 = 2f - f2, (6) p = (x2 + y2)'\ (7) r = (p2 + z2)*, e2a tan u = - (1 f) + (8) P The preceding equations can be used in conjunction with Tables 3.1 and 3.2 superseding the ones previously presented in Soler and Hothera (1988). Table 3.2. Parameters of Some Adopted Reference Ellipsoids. Coordinate system Reference ellipsoid (datum) used a (m) 1/f AGD AN (or SA-69) 6,378,160 298.25 ED-79 International 6,378,388 297 GEM-8 GEM-8 6,378,145 298.255 GEM-9 (or GEM-10) GEM-9 or (GEM-10) 6,378,140 298.255 GEM-10B GEM-10B 6,378,138 298.257 GEM-Tl GEM-Tl 6,378,137 298.257 GEM-T3 GEM-T3 6,378,137 298.257 NAD-27 Clarke 1866 6,378,206.4 294.9786982 NAD-83 GRS-80 6,378,137 298.257222101 NWL-9D = NSWC-9Z2 WGS-66 6,378,145 298.25 SA-69 SA-69 (or AN) 6,378,160 298.25 WGS-72 WGS-72 6,378,135 298.26 WGS-84 WGS-84 6,378,137 298.257223563 Note: AGD = Australian geodetic datum; AN = Australian national; ED = European datum; GEM = Goddard Earth model; GRS = geodetic reference system; NAD = North American datum; NSWC = Naval surface warfare center; NWL = Naval Weapons Laboratory; SA = South American; WGS = World geodetic system. In the case of IERS, these algorithms are used with Table 3.1, where (x,y,z) are coordinates in the specified coordinate system (datum), and (u,v,w,) are identified with ITRF, or a more specific realization (ITRF90 for table 3.1) . In the current practice of the IERS/CB and its publications, (u,v,w) are denoted (X,Y,Z) and the coordinates in an individual system by (XS, YS, ZS) , while the 7 parameters are identified as (T,, T2, T3) refer to (Ax, Ay, Az) , D to 5s and (Ri,R2,R3) to (-Sc, -5*, -5 b) . 19 Transformations to Current Datums Table 3.1 gives values recommended to convert ITRF90 coordi­ nates into other datums. Table 3.2 Lists the required two parameters for several adopted reference ellipsoids defining important geodetic datums. Some ellipsoids were introduced by NASA's GSFC to reference their Goddard Earth Models (GEM, series 8,9, and 10). They were primarily used to obtain geoid heights (undulations) and depict global geoid maps. These transformation formulae should be used with care. There are several ways to determine such parameters, either by direct comparison between two realizations of the datums, or by combining formula through an intermediate datum. This is well illustrated by WGS84 as mentioned below. The numbers given in Table 3.1 have been determined using the following data: a) The transformation between BTS87 and WGS84 derived from Boucher, Altamimi, and Willis (1988). b) Transformation parameters between the Doppler realized frames (i.e., NWL-9D = NSWC-9Z2, WGS-72, WGS-84) have been adopted by the Defense Mapping Agency. Recall that Cartesian coordinates, derived from using the Global Positioning System (GPS), are also referred to the WGS-84 coordinate system. Nevertheless, the Doppler WGS-84 (GPS) frames are not neces- sarily coincident (Malys, 1988). Similarly, although by definition, the NAD-83 and WGS-84 realized Cartesian frames should coincide, small differences (<0.5 ppm) in shifts, rotations, and scale between the two frames may be discovered.